Collect like terms, multiply out single brackets, then pairs of brackets, and (Extended) three brackets. Every term inside must be multiplied by every term outside.
To expand, multiply every term in one bracket by every term in the other, keeping each sign with the term after it. Then collect like terms: only terms with exactly the same letters and powers can be added.
Fig. 1: the area of a rectangle (x + 3) by (x + 2) splits into four parts, one for each pair of terms, so (x + 3)(x + 2) = x² + 5x + 6
Section 01
Collect, then one bracket.
In the lab choose One bracket and set the number outside to a negative value, say \(-3\). Watch the sign of every term inside flip. Then switch to Two brackets and see the grid of four products build up.
Like terms have exactly the same letters and powers: \(3a\) and \(-5a\) are like terms; \(x\) and \(x^{2}\) are not; \(ab\) and \(ba\) are.
Collecting: \(2a + 3b + 5a - 9b = 7a - 6b\). Each sign belongs to the term after it.
One bracket: \(k(a + b) = ka + kb\). The term outside multiplies every term inside.
Worked example Simplify (Extended)
Simplify \(2a^{2} + 3ab - 1 + 5a^{2} - 9ab + 4\).
\(a^{2}\) terms: \(2a^{2} + 5a^{2} = 7a^{2}\).
\(ab\) terms: \(3ab - 9ab = -6ab\).
Numbers: \(-1 + 4 = 3\).
Answer: \(7a^{2} - 6ab + 3\)
Worked example Expand a single bracket
Expand \(3x(2x - 4y)\).
\(3x \times 2x = 6x^{2}\).
\(3x \times (-4y) = -12xy\).
Answer: \(6x^{2} - 12xy\)
Worked example Expand and simplify
Expand and simplify \(5(2a + 3) - 2(a - 4)\).
\(5(2a + 3) = 10a + 15\).
The \(-2\) multiplies both terms: \(-2 \times a = -2a\) and \(-2 \times (-4) = +8\).
\(10a + 15 - 2a + 8 = 8a + 23\).
Answer: \(8a + 23\)
Examiner trap
\(-2(a - 4)\) is \(-2a + 8\), not \(-2a - 8\). The minus sign outside multiplies the \(-4\) too, and minus times minus is plus.
Section 02
Two brackets, four products.
Each term in the first bracket multiplies each term in the second, giving four products (Fig. 1). Then collect the two middle terms.
Worked example Two brackets
Expand and simplify \((2x + 1)(x - 4)\).
\(2x \times x = 2x^{2}\) and \(2x \times (-4) = -8x\).
\(1 \times x = x\) and \(1 \times (-4) = -4\).
\(2x^{2} - 8x + x - 4 = 2x^{2} - 7x - 4\).
Answer: \(2x^{2} - 7x - 4\)
Worked example A squared bracket
Expand \((x - 5)^{2}\).
Write it as two brackets: \((x - 5)(x - 5)\).
\(x^{2} - 5x - 5x + 25\).
Collect: \(x^{2} - 10x + 25\).
Answer: \(x^{2} - 10x + 25\)
Worked example Two letters (Extended)
Expand and simplify \((3x + y)(x - 4y)\).
\(3x \times x = 3x^{2}\) and \(3x \times (-4y) = -12xy\).
\(y \times x = xy\) and \(y \times (-4y) = -4y^{2}\).
\(-12xy + xy = -11xy\).
Answer: \(3x^{2} - 11xy - 4y^{2}\)
Your turn
Expand and simplify \((2x - 3)(3x + 2)\).
\(6x^{2} + 4x - 9x - 6 = 6x^{2} - 5x - 6\).
Examiner trap
\((x + 3)^{2}\) is NOT \(x^{2} + 9\). Write it as \((x + 3)(x + 3)\) and you get the two middle terms: \(x^{2} + 6x + 9\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Expand \(4(x - 3)\).
Quick check+10 XP
Simplify \(5a - 2b - 3a + 6b\).
Quick check+10 XP
Expand \((x + 3)^{2}\).
Quick check+10 XP
Expand \(-3(2y - 5)\).
Section 03
Three brackets (Extended).
Expand two of the brackets first, simplify, then multiply the result by the third bracket. Tick off each product as you go: a three-term bracket times a two-term bracket gives six products.